Hydrodynamics and Elasticity: Class 8
Viscous flows I
1. A laminar river A film of incompressible, viscous fluid of constant thickness h is bounded from above by a free surface and from belowe by a planar surface inclined at an angle α to the horizonta. The flim flows steadily under the influence of gravity down the slope. Determine the velocity profile in the fluid, the maximal and average velocity, and then find its value for water (ν = 0.01 cm2/s) in a channel of length l, height difference H and depth h if
(a) L = 100 m, H = 1 m, h = 0.5 cm (a rivulet)
(b) L = 100 km, H = 300 m, h = 5 m (a simple model of the Vistula river)
In the latter case, how fast would it take to get from Warsaw to Toruń (220 km) in a raft?
2. Taylor-Couette flow Determine the velocity distribution for a fluid contained between two coaxial cylinders, rotating around their axes with angular velocities Ω1 and Ω2. The radii of the cylinders are R1 and R2
(R2> R2). Assume the flow to only have the azimuthal component
vϕ= v(r), vr= vz= 0; p = p(r), (1)
in cylindrical coordinates (r, ϕ, z). Determine the frictional torque acting on the cylinders. How can this flow be used to determine the viscosity of the fluid?
Is it possible to select the boundary conditions in such a way that the flow is irrotational?
(G. G. Stokes, 1845)
3. Slip boundary condition A viscouis, incompressible fluid is flowing in a circular pipe of radius R under the action of a constant pressure gradient −G parallel to the pipe axis. In some circumstances, the no-slip boundary condition does not hold on the pipe surface, hence we replace it with a constant slip boundary condition
u = −λ∂u
∂n at r = R. (2)
Find the velocity distribution in the fluid. Is the result consistent with a no-slip velocity field in the limit of λ → 0? Find the average flow velocity and compare it to the no-slip boundary result.
